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I don't know how you managed to parse it the way you did, but the first sentence in Article 2 clearly says that if you're party to the Conventions, it doesn't matter if your opponent is or not.
No, the first sentence says that if there are multiple parties in the conflict and one or more of them is not party to the GCs, then those that are party to the GCs will remain bound to each other in their mutual relations.
"Although one of the Powers in conflict may not be a party to the present Convention, the Powers who are parties thereto shall remain bound by it in their mutual relations."
So, if parties X, Y and Z are at war and X, Y are signatories to the GCs but Z is not, then X and Y are bound by the GCs in relation to each other (but not with Z). The following sentence then says that even if Z is not a signatory, if Z accepts and implements the tenets of the GCs, then X and Y are also bound by the GCs in their relations with Z.
If the signatories were always bound by the GCs, then why wouldn't they just say exactly that? Why have a whole complicated paragraph discussing what happens when a non-signatory is a party to a conflict?
The HCs also have similar escape clauses because no one would have signed them if it meant that they had to fight with a hand tied behind their back while their enemy did as they liked.